214 Magnetohydrodynamics
9. MAGNETOHYDRODYNAMIC
GENERATION
Chapter IV
(MUD) POWER
When an electrical conductor is moved so as to cut lines of magnetic
induction, the charged particles in the conductor experience a force in a
direction mutually perpendicular to the B field and to the velocity of the
conductor. The negative charges tend to move in one direction, and the
positive charges in the opposite direction. This induced electric field, or
motional emf, provides the basis for converting mechanical energy into
electrical energy. At the present time nearly all electrical power generators
~tilize a solid conductor which is caused to rotate between the poles of a
magnet. In the case of hydroelectric generators, the energy required to
maintain the rotation is supplied by the gravitational motion of river water.
Turbogenerators, on the other hand, generally operate using a high-speed
flow of steam or other gas. The heat source required to produce the.
high-speedgas flow may be supplied by the combustion of a fossil fuel or by
a nuclear reactor (either fission or possibly fusion).
It was recognized by Faraday as early as 1831 that one could employ a
fluid conductor as the working substancein a power generator. To test this
concept Faraday immersed electrodesinto the Thames river at either end of
the Waterloo Bridge in London and connected the electrodesat mid span on
the bridge through a galvanometer. Faraday reasoned that the electrically
conducting river water moving through the earth's magnetic field should
produce a transverse emf. Small irregular deflections of the galvanometer
were in fact observed.The production of electrical power through the useof a
conducting fluid moving through a magnetic field is referred to as magnetohydrodynamic, or MHO, power generation. One of the earliest serious
attempts to construct an experimental MHO generator wasundertaken at the
Westinghouse laboratories in the .period 1938-1944,under the guidance of
Karlovitz (seeKarlovitz and Halasz, 1964).This generator (which was of the
annular Hall type-see Fig. 20) utilized the products of combustion of
natural gas, as a working fluid, and electron beam ionization. The experiments did not produce the expected power levels because of the low
electrical conductivity of the -gas and the lack of existing knowledge of
plasma properties at that time. A later experiment at Westinghouseby Way,
OeCorso, Hundstad, Kemeny, Stewart, and Young (1961),utilizing a liquid
fossil fuel .. seeded" with a potassium compound, was much more successful
and yielded power levels in excessof 10 kW. Similar power levels were
achieved at the Avco Everett laboratories by Rosa (1961) using arc-heated
argon at 30OQoK.. seeded" with powdered potassium carbonate. In these
latter experiments .. seeding" the working gas with small concentrations of
potassium was essential to provide the necessarynumber of free electrons
Icc"cC~ic'!i!,,"
1
'"';!!,~
Section 9
MUD Power Generation
215
required for an adequate electrical conductivity. (Other possible seeding
materials having a relatively low ionization potential are the alkali metals
cesium or rubidium.)
During the decade beginning about 1960 three general types of MHD
generator systemsenvolved, classified according to the working fluid and the
anticipated heat source. Open-cycle MHD generators operating with the
products of combustion of a fossil fuel are closest to practical realization.
In the United States,operation ora 32 MW alcohol-fueled generator with run
times up to three minutes was achieved in 1965 (seeMattsson, Ducharme,
Govoni, Morrow, and Brogan, 1965). In the Soviet Union tests on a
75 MW (25 MW from MHD and 50 MW from steam) pilot plant burning
natural gas began in 1971. Closed-cycle MHD generators are usually
envisaged as operating with nuclear reactor heat sources, although fossil
fuel heat sourceshave also been considered.The working fluid for a closedcycle systemcan be either a seedednoble gas or a liquid metal. Becauseof
temperature limitations imposed by the nuclear fuel materials used in
reactors, closed-cycleMHD generators utilizing a gas will require that the
generator operate in a nonequilibrium mode. We shall have more to say later
about some of the difficulties that nonequilibrium operation entails. The
subject of liquid metal MHD generators lies outside the scope of our
discussion.
An MHD generator, like a turbogenerator, is an energy conversion device
and can be usedwith any high-temperature heat source-chemical, nuclear,
solar, etc. The future electrical power needs of industrial countries will
have to be met for the most part by thermal systemscomposed of a heat
source and an energyconversion device.In accordancewith thermodynamic
considerations, the maximum potential efficiency of such a system (i.e., the
Carnot efficiency) is determined by the temperature of the heat source. However, the maximum actual efficiency of the system will be limited by the
maximum temperature employed in the energyconversion device.The closer
the temperature of the working fluid in the energy conversion device to the
temperature of the heat source, the higher the maximum potential efficiency
of the overall system.A spectrum of heat source temperatures are currently
available, up to about 3000oK. However, at the present time large centralstation power production is limited to the use of a single energy-conversion
scheme-the steam turbogenerator-which is capable of operating economically at a maximum temperature of only 850oK.The over-all efficiencies
of present central-station power-producing systems are limited by this
fact to values below about 42 percent, which is a fraction of the potential
efficiency. It is clear that a temperature gap exists in our energy conversion
technology.
BecauseMHD power generators, in contrast to turbines, do not require
216 Magnetohydrodynamics
ChapterIV
the use of moving solid materials in the gas stream, they can operate at
much higher temperatures. Calculations show that fossil-fueled MHD
generators may be capable of operating at efficiencies between 50 and 60
percent.Higher operating efficiencieswould lead to improved conservation of
natural resources,reduced thermal pollution, and lower fuel costs. Studies
currently in progress suggestalso the possibility of reduced air pollution.
In this section an elementary account of some of the concepts involved in
MHD power generation is presented. A more complete discussion may be
found in the book by Rosa (1968).
The essentialelementsof a simplified MHD generator are shown in Fig.
15. This type of generator is referred to as a continuous electrode Faraday
generator. A field of magnetic induction B is applied transverse to the
):.
Ionized gas
source",
z
x
Figure 15. A simplified MHO generator.
I
I
:
--""_"""""eu.,"
""""""""""c'"
""",~"'~~;;"~"ccC'-,cc,
-
Section9
MUD Power Generation 217
motion of an electrically conducting gas flowing in an insulated duct with a
velocity u. Charged particles moving with the gas will experiencean induced
electric field u x B which will tend to drive an electric current in the
direction perpendicular to both u and B. This current is collected by a pair
of electrodes on opposite sides of the duct in contact with the gas and
connected externally through a load. Neglecting the Hall effect, the
magnitude of the current density for a weakly ionized gas is given by the
generalized Ohm's law (8.26) as
J = u(E + u x B).
(9.1a)
The electric field E, which is added to the induced field, results from the
potential difference between the electrodes. For the purposes of our initial
discussion in this section we shall assume that both u and u are uniform.
In terms of the coordinate system shown in Fig. 15, we have that
Jy=u(Ey-uB).
(9.1b)
At open circuit J y = 0, and so the open circuit electric field is uB
[cf. equation (7.3)]. For the characteristic conditions u 1000 m sec-1 and
B 2 T, the open circuit electric field is uB"'"' 2000 V m-1. At short
circuit Ey = 0, and the short circuit current is J y = - uuB. For general
load conditions, it is conventional to introduce the loading parameter
"'"'
"'"'
Ey
K=- -,
uB
(9.2)
where 0 ~ K ~ 1,and write Jy = -uuB(1 - K). The negative sign indicates
that the conventional current flows in the negative y-direction. Since the
electrons flow in the opposite direction, the bottom electrode must serveas an
electron emitter, or cathode, and the upper electrode is an anode.
In accordance with equation (4.5a), the electrical power delivered to the
load per unit volume of a MHD generator gas is
P = -J.
E.
(9.3a)
For the generator shown in Fig. 15,
P = uu2B2K(1 - K).
(9.3b)
This power density has a maximum value
Pmax
.
B2
= UU2
-~'
(9.4)
for K = 1/2. In accordancewith equation (4.4),the rate at which directed
energy is extracted from the gas by the electromagneticfield per unit
~--j~,""~~
'" ',","cC'"C""'",
218 Magnetohydrodynamics
Chapter IV
volume is -00 (J x B). We therefore define the electrical efficiency of a
MHO generator as
JoE
l1e
=0
0
(J x B) .
(9.5a)
For the generator being discussed,
l1e= K.
(9.5b)
The Faraday generator therefore tends to higher efficiency near open circuit
operation.
In order that a MHO generator have an acceptable size, it is necessary
that the generator deliver a minimum of about 10 MW per cubic meter
of gas.Using the precedingcharacteristic valuesfor u and B, this requirement
means that the electrical conductivity must be such that
0" ~ 4 2p Max
2"'"
10 mhos m - 1 .
(96)
.
uB
102
-
/"" ,
E
0
.c
E
>
I
""
.Co)
~I:
.> 10
...
/
,
/
;'
-IV
,
,
I
I
..
,,
',
,
10-1
1600
..
'"
.."
Co)
1
--
'"
,,
.~
...
""
",/
'
0
Figure 16.
,
/
...
w
--' ---
.
I
...
Co)
~
-
-.
-.
--~
----
,
,,
1800
,,
,
..
,'
","
/
,
2000
2200
2400
Temperature, 0 K
2600
2800
3000
Representative values of the electrical conductivity ofMHD generator
plasmas at 1 atm
(-
products of combustion
of C 2HsOH + 302 with 0.5 mass
fraction N2/O2, seededwith 0.01 mass fraction K; --argon seededwith
0.004 mole fraction K;
argon seededwith 0.004mole fraction Cs).
cc__cc,,\,'-
C_""""",~c
c
"'V"-*"'ccc~~=~~c,,"
Section9
MUD Power Generation 219
Current densitieswill then be of the order of a few amperes per square cm
or more. The equilibrium electrical conductivities at atmospheric pressure
of a potassium-seededcombustion products plasma, and potassium-seeded
and cesium-seededargon plasmasare shown in Fig. 16, plotted as a function
of temperature. The gas temperatures needed to achieve the condition (9.6)
can be readily obtained with many fossil fuels. However, exit gas temperatures available from present-day nuclear reactors are too low, and it is
necessaryto examine the feasibility of employing nonequilibrium methods to
obtain enhanced values of the electrical conductivity.
Shown in Fig. 17 are the values of the electron Hall parameter for a
magnetic induction of 1 T, corresponding to the gases and conditions
described in Fig. 16. (To a good approximation, the Hall parameter scales
linearly with B.) It is apparent that the Hall effect can playa significant
role in the operation of an MHD generator, and it is necessaryto review
the preceding analysis. Because of the Hall effect, a current flowing in
the y-direction can give rise to a current flowing in the x-direction. Instead
7
6
".
'.
"
"
'"
--
~
.-". "'...
"
5
...
Q)
.
...
'"
,",
.,
""",
"
'" .,
",
E
IU
~4
-
.,',.'"'
-IU
,
" .,
:I:
c:
e3
u
-UJQ)
,,'-
,
'-,
'
,.
""-.
2
'
1
0
1600
1800
2000
2200
2400
Temperature, 0 K
2600
2800
3000
Figure 17. Representativevalues of the electron Hall parameter of MHO
= 1.0 T at 1 atm (products of combustion of
C2HsOH + 302 with 0.5 massfraction N2/O2, seededwith 0.01massfraction K;
--argon seededwith 0.004 mole fraction K;
argon seededwith
0.004 mole fraction Cs).
generator plasmas for B
»,,"..
""~,
220 Magnetohydrodynamics
Chapter IV
of the Ohm's law in the form of equation (9.1), we must now write
[cf. equations (8.26), (3.21), and (3.11)]
=
~
1+"
(E~+ PEx),
(9.7a)
Jx =
~
(Ex - PE~).
(9.7b)
Jy
and
(For simplicity, we shall employ the notation P = Pehenceforth.)Because
the electrodesin a generator of the type shown in Fig. 15 extend the entire
length of the duct, they tend to impose equipotential surfaces in the gas
which are normal to the y-direction. Thus for a continuouselectrodeFaraday
generator
Ex = O.
(9.8a)
It follows from equations (9.7) that
.
J = ~E' = -O'uB(l - K)
y
1 + p2 y
1 + p2
'
Jx =
- pO'
i+p2
,
Ey
= -Ply,
(9.8b)
(9.8c)
and from equation (9.3a) that
p
O'u2B2K(1 = i--:+p2
K).
(9.8d)
The Hall effect reduces J y and P by the factor (1 + P2) and results in the
appearance of a Hall current which flows downstream in the gas and
returns upstream through the electrodes. The reduction of J y and P is caused
by the fact that the uB field must overcome not only the Ey field produced
by the electrodes,but also the Hall emf resulting from the current flow in the
x-direction.
To circumvent the deleterious consequences of the Hall effect, the
electrodes may be segmented in the manner indicated in Fig. 18b and
separate loads connected between opposed electrode pairs. In the limit of
infinitely fine segmentation, there can be no x-component of current either
in the electrode.4 or in the gas, and so the condition for an ideal
segmentedFaraday generator is
J x = O.
~"
(9.9a)
'cc~c~
~
,."'"
1
cc.~
Section9
MUD Power Generation 221
z
y
x
:::-""U
(0) Continuous Faraday
(b) Segmented Faraday
(c) Hall
(d) Diagonal conducting wall
Figure 18.
Electrode connections
for linear MHD
generators.
From equation (9.7b) we see that this configuration leads to the build-up
of an electric field in the x-direction, the so-called Hall field
(9.9b)
Ex = pE~.
With this value for Ex, equation (9.7a) becomes
Jy = (J'E~,
(9.9c)
which is identical to the relation (9.1b) that we used when we neglected the
Hall effect. We also recover the value for the power density in the
~
""""'~C""
"'""""'"
jiJ",""~~,"",t
""""
222 Magnetohydrodynamics
Chapter IV
absenceof the Hall effect given by equation (9.3b). In an actual generator
with finite segmentation, the Hall effect causesthe current to concentrate
at the upstream edgeof an anode and at the downstream edge of a cathode
(see Rosa, 1968, p. 68). Thus the Hall currents are not completely suppressed, and the improvement in performance of a segmented Faraday
generator is not as good as the foregoing idealized calculation would suggest.
An experimental power curve for a combustion products MHD generator
obtained by Way, De Corso, Hunstad, Kemeny, and Stewart (1961)is shown
in comparison with theory, in Fig. 19.The discrepancy is attributed in part
to leakage currents in the generator walls.
14
12
10
~
~
~-
~
8
6
0
Q..
4
2
0
100
200
300
400
Current, A
Figure 19. Electric power produced by an experimental combustion products
MHO generator compared with theory (after Way. DeCorso, Hundstad, Kemeny,
Stewart, and Young, 1961).
The action of the Hall effect in building up an axial electric field
suggeststhat this field can be used to drive a load connected between the
upstream and downstream electrodes. According to equation (9.9b), the
magnitude of the axial electric field will be a maximum when the opposed
electrodes are shorted, i.e.,
E" = o.
(9.10a)
In principle the shorting may be accomplishedeither externally or internally,
as indicated in Fig. 18c. This type of connection is referred to as a Hall
generator. For the condition (9.10a) the Ohm's law equations (9.7) become
J"
=
(1'
1+
p2(-uB
+ PEx),
(9.lOb)
and
Jx =
(1'
l+P
2
(Ex+ puB).
(9.1Oc)
j
~_.
.c"~"""~
~
Section 9
MUD Power Generation
223
The open-circuit electric field for a Hall generator is obtained from
equation (9.1Oc)as -puB. We may therefore define the loading parameter
for a Hall generator to be
KH
=~ '
-E
(9.10d)
f'uB
and rewrite the Ohm's law equations in the form
Jy =
-
O"uB(1 + P2KH)
2'
(9.10e)
l+P
and
p (1 - KH).
J x = O"uB["+Ill
(9.101)
For a Hall generator,the power densityis
~
KH(l - KH),
P = -JxEx = O"u2B2
(9.10g)
and the electrical efficiency is
Jx Ex = 1 +p2
.
1le= J;UB
p2K~
KH( 1 - KH).
(9.10h)
For large Hall parameters,the power density of a Hall generator approaches
that of a segmentedFaraday generator. In contrast to segmentedFaraday
generators, Hall generators tend to have higher electrical efficiencies near
(but not at) short circuit and otTerthe simplicity of a two-terminal connection.
The continuous Faraday and Hall generators may be viewed as special
casesof a classof two-terminal generators referred to as diagonal conductingwall generators. As illustrated in Fig. 18d, the diagonal conductors, which
form part of the duct, are in contact with the plasma and tend to impose
diagonal equipotential surfacesin the plasma. Thus, the condition defining a
particular diagonal conducting wall generator is
Ey -tan
E=
(9.11)
lX,
x
where lx is the angle between the plane passing through a conductor and the
y-z plane. The Hall and continuous Faraday generators correspond respectively to selecting lx as either 0 or n/2. For further discussion of these
generators, the reader is referred to the book by Rosa (1968).
In addition to the linear geometry and its various electrodeconfigurations,
MHD generators having other geometries have also been proposed. In
-..l
~.Ji
"~~
224 Magnetohydrodynamics
Chapter IV
the disk and annular Hall generators shown in Fig. 20, the current component induced to flow in the u x B direction closes upon itself within the
plasma, thereby obviating the need for segmentedelectrodes. The vortex
geometry is similar to a continuous electrode Faraday generator and thus
requires that the Hall parameter be small.
It has beenpointed out by Rosa (1962)that nonuniform plasma properties
in high Hall parameter devicescan causelarge degradations in performance.
To show this effect let us consider a MHD duct with either continuous or
finely segmentedelectrode walls, in which 0' and fJ depend only on the coordinate y. Sucha nonuniformity could result, for example,from wall-cooling,
from insufficient seed-mixing, from nonuniform combustion, or from
nonequilibrium effects. Let us assume that a steady-state solution of the
electrodynamic equations exists where J and E' also depend on the ycoordinate only. It then follows from the equations V. J = 0 and
V x E = 0 [cf. equations (6.10e)and (6.101)],that
J, = constant, Ex = constant.
(9.12)
The overall performance of the device will depend on the values of J x(y)
and E~(y) averaged over the y dimension of the duct. Solving the Ohm's
law equations (9.7) for the latter quantities in terms of the former and
averaging the result, we obtain
-
E~=
( 1;-+ fJ2) J'-fJEx'
(9.13a)
lx = uEx - pI,.
(9.13b)
If h denotes the y-dimension of the duct, then for any quantity f(y),
1== h-1 J~f(y) dy. Equations (9.13)expressthe averagedfield quantities
in terms of averagedplasmaproperties.The simplicity of this result stems
from the assumedone dimensionalityof the nonuniformity.
If we recast equations (9.13)into the form of the original Ohm's law
equations,we obtain
( --;1 + fJ2) J, = -,E, + fJEx,
(9.14a)
( 1;-+ fJ2) -Jx = GEx- -,
fJE,.
(9.14b)
The nonuniformity factor
G = if(~)
- p2
(9.15a)
:,.
~~;.w -~
~"
-",
u
u
u
Disk Hall
B
Electrode
Annular Hall
B
Exhaust
Gas inlet
Vortex
Figure 20. MHD generator geometries.
~~,"-
~","';j"
226 Magnetohydrodynamics
Chapter IV
contains the coupling of nonuniformities and the Hall effect and is equal
to unity for a uniform gas. If the Hall parameter is uniform,
G = 1 + (1 + P2)(UU-=1
- 1).
(9.15b)
This expressionshows how the effect of even a small departure from a uniform conductivity becomesamplified at large Hall parameters.For an ideal
segmentedFaraday generator, we obtain in place of equations (9.9b) and
(9.3b),
Ex
= ~/IE ,
(9.16a)
G
and
2P = --2
C1U B K(1 - K),
(9.16b)
G
where K = Ey/(uB). Thus, both the Hall field and the power density are
reduced by the factor 1/G. The dependenceof 1/G on p and on the degree
of nonuniformity is shown in Fig. 21 for a particular distribution of C1(y).
The elevation of electron temperature in a uniform and steady discharge
in a noble gas seededwith an alkali metal is given by the electron energy
equation [cf. equation (5.4)] as
J. E' = 3nek(Te- T) ~
veH
+ R.
(9.17)
mH
1.0
h
0.8
Y
o~2
0
uwao
a(y)
0.6
1
G
0.4
0.2
0
10-3
10-2
10-1
aw/uo
1
Figure 21. Effect of a nonuniform conductivity profile on the power density
of a segmented Faraday MHD generator.
i
I
CC"._C"~"""c+
CCC~"c\.
-
::..".'""'.!-
Section 9
MUD Power Generation
227
Here mHis a weighted average of the massesof the heavy particles, T is the
heavy-particle gas temperature, .k is the local net rate of radiation energy
loss per unit volume, and we have employed the approximation Je ~ J.
For a dischargewith an applied electric field and zero B field, the left-hand
side of this equation can be written P /0". If we assume that ne and Te are
related by Saha's equation [cf. equations (II 10.5)], and if we use the
relation
0" = nee2/meVeH' then equation (9.17) may be used to predict the
dependenceof 0"on the current density J. Experiments to test this model
at atmospheric pressuresand high gas temperatures were first undertaken
by Kerrebrock (1962). Some representative experimental results and a
comparison with theory obtained by Cool and Zukoski (1966)are shown in
Fig. 22 for potassium-seededatmospheric pressureargon. The interpretation
of the data at low current densities is complicated becauseof frozen-flow
.
and other nonequilibrium effects,but at current densities of practical
interest agreement between theory and experiment is quite satisfactory.
These experiments show that the electrical conductivity can be significantly
increased by nonequilibrium ionization.
For a MHD generator, the left-hand side of equation (9.17) may be
evaluated using the Ohm's law relations (9.7). Neglecting the radiation
loss term, we obtain
3 kT( Te
) me- - 1 -VeH
T
mH
ne
=
O"E,2
2'
1 + fJ
(9.18)
The value of E,2 depends on the type of MHD generator being considered.
Shown in Table 2 are expressionsfor the ratio of the electron temperature
to the gas stagnation temperature
To = T[l
+ YM2],
for each of the three main types of linear MHD generator previously
discussed.Here y is the ratio of specific heats of the gas, y = (RmH/k)y~ y,
and
2
M2=~
yRT
is the square of the Mach number of the gas.To make more evident the dependenceon the Hall parameter, the last line of Table 2 shows the limiting
values of Te/T0 for y
=y=
5/3, large Mach number, and optimal loading. The
maximum elevation in electron temperature for a continuous Faraday
generator is of the order of the gas stagnation temperature. However, for
,
I
""-
C""'""~~
10
Theory and data for argon-potassium
T = 2000oK,(nK/nA) = 0.004
E
--uIn
0
.c
E
.~
1.0
Elastic collision losses
>
':;;' 0.7
u
~
"0
c:
0
u
0.5
0.4
-
0.3
.g
0.2
...
u
~
w
Elastic collision losses and radiation
0.1
0.07
0.05
0.040.02
.
.
.
.5.0 1
2
5
10
20
22
24
50
100
Current density, A/cm2
(a) Electrical conductivity
3600
3400
~
0
3200
- 3000
ft
GO
...
~
l?:
GO
2800
~
2600
a.
...
c:
.2
...
2400
~
~
a.
0
2200
Q..
2000
1800
1600
0
12
14
18
20
26
28
Current density, A/cm2
(b) Electron temperature
Figure 22. Nonequilibrium
ionization in atmospheric pressure potassium-seeded
argon (after Cool and Zukoski, 1966).
"~
c;" "
-,
"c.,"...~~
j
30
Section9
MHD Power Generation 229
Table 2
Magnetically Induced Elevation in Electron Temperature for Closed
Cycle Linear MUD Generators
Faraday
Ex
E~
Continuous
Segmented
0
-(1 - K)uB
-(1 - K)fJuB
-(1 - K)uB
1 + 1(1- K)2
3
-Te
To
M» 1
5}
Y = "3
~
M2
1+
5 fJ2
3- 1 + fJ2'. K
1+
~
1(1+ fJ2K:')
3
3
1 + !!-=-!l M2
2
0
-+
KHfJuB
-uB
1(1- K)2 fJ2M2
1 + fJ2
1 + !!-=-!l M2
2
-+
Hall
-+
-5 fJ2. K
3
'
-+
M2
1 + fJ2
1+ ~
M2
2
0
-+
()
-53 1 +fJ2fJ2 fJ2; K H -+
1
the segmentedFaraday and Hall generators, this theory predicts that considerably higher electron temperatures are possible at high values of the
Hall parameter.
The enhanced electrical conductivities implied by the preceding theory
have not been experimentally observed. The reasons are not completely
understood at the present time, but it is generally believed that a major
contributing factor is that the plasma becomesunstable and that fluctuating
inhomogeneities in the plasma properties develop. We shall discuss the
origin of this instability in the next section.
It has been suggestedby Solbesand Kerrebrock (1968) that for a Faraday
generator, one should write in place of the expressions in Table 2, the
relation
~T.
1 = r M2(1 -
K)2p2
3
(~ )
0'
~_.:!_~~~.
1 + Pelf
(9.19)
The bar here denotes a spatial average; the effective Hall parameter Pelf is
defined as the tangent of the angle between J and F (see Fig. 9); the
effective conductivity is defined by the relation O'elf==J/~, where ~ is the
average of the projection of E' on J; the apparent Hall parameter is
defined by the equation
Papp=='E:/[uB(l
- K)]. For a uniform plasma
equation (9.19) reduces to
~T -
r
1 = 3 M2 (1 - K )2P2 ~~_&
1 + p2 '
230 Magnetohydrodynamics
Chapter IV
from which one may see the extreme sensitivity, for large p, of Te/T to
the Hall voltagerecoveryPapp/P. For an unstableplasmaPeff is lessthan II,
and appears to saturate at a value of order unity for large P, while
O'eff
appears (in some experiments) to approach PetriP.
Thus, for an unstable
plasma (~/T) - 1 varies as II for large p. This result should be contrasted
with the behavior ofa stable plasma for large P, where (Te/T) - 1 variesas
p2 for segmented electrodes, but is independent of Pfor continuous electrodes.
It would appear on the basis of expression (9.19) that nonequilibrium
operation of a Faraday generator should be possible even in the presenceof
instabilities and that the level of nonequilibrium should increase with
increasing magnetic field, albeit not as strongly as predicted for an ideally
segmentedFaraday generator.
Exercise 9.1. For a Hall generator, show that the loading parameter for
maximum
electrical efficiency is KH = (.)1-+112- 1)/P2, and that the
maximum electrical efficiency is "Ie= 1 - 2(.Jl-""+Ji2- 1)/P2.
Exercise9.2. Discuss the performance featuresof a diagonal conducting
wall generator.
Exercise 9.3. Discuss the effects of one-dimensional nonuniformities on
the performance of a Hall generator.
Exercise 9.4. Compare the effects of ion slip on the maximum power
outputs of the three main types of MHD generator electrode connections.